19.[命题立意]考查线面垂直,面面垂直,二面角;考查直观想象,逻辑推理,数学运算的核心素养[试题解析](1)证明:因为PD⊥底面ABCD,BCC底面ABCD,所以PD⊥BC又CD⊥BC,PD∩CD=D,PD,CDC平面PCD,所以BC⊥平面PCD(1分)又DMC平面PCD,所以DM⊥BC(2分)因为底面ABCD为正方形,所以AD=CD所以在△PCD中,PD=CD因为M为PC的中点,所以DM⊥PC(3分)又PC∩BC=C,PC,BCC平面PBC所以DM⊥平面PBC,(4分)又DMC平面DNM,所以平面DNM⊥平面PBC(5分)(2)以D为坐标原点,AD,DC,DP所在直线分别为x轴、y轴、z轴,建立如图所示的空间直角坐M标系,设PD=1,N(A,10),则A(1,0,0),B(1,1,0),P(0,0,1),M(0,1Dy1),所以A下=(-1,0,x1),AB=(0,1,0),D=(,1,0),D=(0,2,2)(6分)(6分)设平面PAB的法向量为m=(x1,3y1,z1则m·AB=0,即x1+z1=0,y1=0令x1=1,可得m=(1,0,1),(8分)设平面DNM的法向量为n=(x2,y2,22),Ax:+yu022n·D=0,少+令x2=1,可得n=(1,-A,A),因为平面DNM与平面PAB的夹角为30,(10分)1+入2,解得即所以×√1所以当N为线段BC的中点时,平面DNM与平面PAB所成锐二面角的大小为30°(12分)椭圆的方程,抛物线的方程,直线与椭

【语篇解读】这是一篇记叙文,讲述了 Sparkes从为了孩子而减肥到自觉健身再到跑马拉松的故事24.A【解析】细节理解题。根据文章第二段中的" His doctor was actually shocked that he didn' t have anyunderlying health issues..可知,医生对 Sparkes没有健康问题而感到惊讶。故选A。25.D【解析】细节理解题。根据第三段中的" The real motivation to change came when the3l- year-old recognizedthat he didn't have the energy to play with his two young children."可知, Sparkes是一个“负责任,有担当”的父亲。故选D。26.A【解析】推理判断题。根据第四段中的描述“…, supportive… worried… no stress or strain with themarriage. loves the weight loss:"可知,他们的婚姻美满。故选A。27B【解析】段落大意题。根据最后一段的内容,尤其是“…he' s achieved his fitness goals and lost the weight,,Sparkes is looking into reconstructive surgery to remove his excess skin. After 2021 he also plans to begintraining for his first ultramarathon, a 50-mile race that only the most hardcore runners ever attempt. " HJ知, Sparkes持有“积极,上进”的生活态度,由此可知B项正确。

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